claimroutineinferred_dep

Fixed exponents satisfy all parameter conditions

Chosen rational exponents satisfy all layer and dimension inequalities

admissible parameters for the layer and cost table

Claim·§9.1 A fixed rational choice·inside thmInteger multiplication in O(n (lg n)^{1-κ}) time

Statement (paper)

For the fixed rational numbers in , the layer conditions hold:

\max\{\tau,\sigma\}<\lambda<1

\tau(1+c/\beta)<1-31\cdot2^{-55}<\lambda

, and

\sigma+\beta(1-\sigma)=1-2^{-51}<\lambda<\lambda'<1

. The dimension-exponent conditions also hold:

\epsilon<\frac1{12}

\epsilon C_1<1

\epsilon(2-\tau)<1-\tau

\frac34+\delta+\frac32\epsilon<1

\epsilon(1+c)<1

, and

\epsilon+\delta<1

In plain English

With the specific rational parameter values fixed in the paper, every required inequality holds: the layer conditions (such as

\max\{\tau,\sigma\}<\lambda<1

and

\sigma+\beta(1-\sigma)=1-2^{-51}<\lambda<\lambda'<1

) and the dimension-exponent conditions (such as

\epsilon<\frac1{12}

\epsilon C_1<1

, and

\epsilon+\delta<1

Why it's here

This is a finite arithmetic check that the chosen rational constants are admissible, so the simultaneous Hadamard layer proposition and later band-bound and cost-table claims can be applied with these parameters. It separates the bookkeeping of exponent inequalities from the main argument, and in particular confirms

\epsilon C_1<1

, which makes the guard width

o(p)

Classified routine:

A finite arithmetic verification that the paper's chosen rational constants satisfy the inequalities required by the simultaneous Hadamard layer proposition and the later dimension-exponent conditions. It is pure bookkeeping with no new idea.

Selected node

Statement (paper)

In plain English

Why it's here